Section 5.1
1a) zeros are x = 3 multiplicity even x = -1 multiplicity odd
1b) touches the x-axis at x = 3, crosses the x-axis at x = -1
1c) has at most 2 turning points
1d) Will behave like g(x) = x3
1e) falls to the left, rises to the right
1f) sketch a graph and approximate the turning points, also label the x-intercepts
1g) increasing (−∞, .33) ∪ (3, ∞) decreasing (.33, 3)
3a) zeros are x = 3/2 has even multiplicity and x = 1 odd multiplicity
3b) graph touches x-axis at x = 3/2 and crosses x-axis at x = 1
3c) has at most 2 turning points
3d) g(x) = 4x3
3e) falls to the left, rises to the right
3f) window used x-min -3 x-max 3 ymin -1 ymax 1
Section 5.1 continued3g) increasing (−∞, 1.17) ∪ (1.5, ∞)
decreasing (1.17, 1.5)
Section 5.1 continued
5a) 𝑥 = √3 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑥 = −√3 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑥 =
5b) graph touches x-axis at √3 𝑎𝑛𝑑 − √3 𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑥 − 𝑎𝑥𝑖𝑠 𝑎𝑡 𝑥 =
−1
2
ℎ𝑎𝑠 𝑜𝑑𝑑 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦
−1
2
5c) will have at most 4 turning points
5d) will behave like g(x) = 2x5
5e) Falls to the left rises to the right
Window x-min -10 x-max 10 y-min – 10 y-max 20
5f)
5g) increasing: (−∞, −√3) ∪ (−1, .6) ∪ (√3, ∞) decreasing (−√3, −1) ∪ (.6, √3)
7a) 𝑥 = √6 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑥 = −√6 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑥 = 6 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦
7b) Graph touches x-axis at all x-intercepts
7c) has at most 5 turning points
7d) will behave like g(x) = 4x6
7e) rises to the left and rises to the right
Window used X min -10 X max 10 Y min -1000 Y max 10000
7f)
7g)
: 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 (−√6, −.45) ∪ (√6, 4.45) ∪ (6, ∞)𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 (−∞, −√6) ∪ (−.45, √6) ∪ (4.45,6)
Section 5.1 continued
9a)
9b)
9c)
9d)
9e)
x = -1/2, even multiplicity
graph touches x-axis at x = -1/2
has at most 3 turning points
will behave like g(x) = 4x4
rises to the left and rises to the right
9f)
9g)
1
1
answer: increasing (− 2 , ∞) decreasing (−∞, − 2)
Section 5.1 continued
11a) x = 6, even multiplicity
11b) Graph touches the x-axis at x = 6
11c) has at most 3 turning points
11d) will behave like g(x) = 4x4
11e) rises to the left, rises to the right
11f)
g) increasing (6, ∞) decreasing (−∞, 6)
13a)
13b)
13c)
13d)
13e)
13f)
x = -7, odd multiplicity
x = 1, odd multiplicity
graph crosses x-axis at x =-7 and at x = 1
Answer has at most 1 turning point.
like g(x) = x2
rise to left and rise to the right
13g) increasing (−3, ∞) decreasing (−∞, −3)
Section 5.1 continued
15a) x = -2 odd multiplicity
x = 2 odd multiplicity
15b) graph crosses x-axis at both x = 2 and x = -2
15c) will have at most 1 turning point.
15d) will behave like g(x) = x2
15e) rises to the left and rises to the right
15f)
15g) increasing (0, ∞) decreasing (−∞, 0)17a) x = 0
multiplicity)
17b) graph will cross x-axis at each 0.
17c) has at most 2 turning points.
17d) will behave like g(x) = -5x3 for large values of x.
17e) rise to the left fall to the right
x = -3/5
x = -2 (all zeros have odd
Section 5.1 continued
17f)
17g) decreasing (−∞, −1.45) ∪ (−.27, ∞) increasing (-1.45, -.27)
19a)
19b)
19c)
19d)
19e)
19f)
x = 0 has even multiplicity, x = -2 and x = 2 have odd
graph touches at 0 and crosses at -2 and crosses at 2
has at most 3 turning points
graph will behave like g(x) = -3x4
falls to the left, falls to the right
19g) increasing (−∞, −1.41) ∪ (0,1.41) decreasing (-1.41, 0) (1.41, ∞)
Section 5.1 continued
21a)x = 0 has even multiplicity, x = -3/5 has odd and x = - 2 has odd
21b) graph touches x-axis at x = 0 and crosses x-axis at x = -3/5 and crosses at x = -2
211c) it has at most 3 turning points
21d) graph will behave like g(x) = 5x4
21e) rise to left and rise to the right
21f) sketch a graph and approximate the turning points, also label the x-intercepts
21g) increasing (-1.57, - .38) ∪ (0, ∞) decreasing (−∞, −1.57) ∪ (−.38,0)
23) f(x) = x3 – x2 – 14x + 24
27) f(x) = 3x2 – 8x – 3
25) f(x) = 2x3 -11x2 + 17x – 6
29) f(x) = x2 – 2
31) f(x) = x3 – 5x2 – 2x + 10
Section 5.2
1a) 3x2 – 2x + 5 remainder 0
3a) 4x2 – 9 remainder 0
5a) 3x2 + 2x + 12 remainder 0
7a) 5x2 – 10x + 26 remainder 44
9a) x2 + 3x + 9 remainder of 0
11a) (-1) is a zero, there are others
13a) 3 is a zero
15a) (-2) and (1) are zeros
17a) (-5) and (1) are zeros
19a) (-1) and (1) are zeros
21) x = -1, 2, ±2𝑖
27) x = -3, -2,
1±𝑖√3
2
1b) 3𝑥 3 − 17𝑥 2 + 15𝑥 − 25 = (x-5)(3x-5)(x+1)
3b) 4𝑥 3 + 8𝑥 2 − 9𝑥 − 18= (x+2)(2x+3)(2x-3)
5b) 3x3 – 16x2 – 72 = (x-6)(3x2 + 2x + 12)
7b) not applicable, remainder is not 0
9b) (x-3)(x2 + 3x + 9)
11b) f(x) = (x+1)(x-2)(x+3)
13b) f(x) = (x-3)(x-3)(2x-1) or (x-3)2(2x-1)
15b) f(x) = (x+2)(x-1)(x-1)(x+1) or (x-1)2(x+2)(x+1)
17b) f(x) = (x+5)(x-1)(2x+3)(x+3)
19b) f(x) = x4 + 7x2 – 8 = (x+1)(x-1)(x2 + 8)
3
23) x = -2, ± 2
25) x = -3, 2, -1, -3/2
29) x = -1, 1, ±2𝑖√2
Section 5.3
1) 𝑓(𝑥) = 𝑥 3 − 8𝑥 2 − 11𝑥 + 148
5) 𝑓(𝑥) = 𝑥 4 − 10𝑥 3 + 30𝑥 2 − 40𝑥 + 104
9) 𝑓(𝑥) = 𝑥 4 + 𝑥 3 − 8𝑥 2 + 4𝑥 − 48
1
3) f(x) = 𝑥 3 − 8𝑥 2 + 22𝑥 − 20
7) 𝑓(𝑥) = 𝑥 4 + 34𝑥 2 + 225
11) x = 2i, -2i, 4
you may write ±2𝑖, 4
13) x = ±2𝑖, 2 , −3
15) x = 5 + 2i, 5 – 2i, -3
17) x = -4+i, -4 – i, 1
19) x = 3+2i, 3-2i, 2
Section 5.4
1a) domain all real numbers except (-3)
1b) vertical asymptote equation x = -3
3a) domain all real numbers except -1 and 4 3b) answer: vertical asymptote x = -1 and x = 4
5a) domain all real numbers except 2, -2
5b) vertical asymptote x = 2 and x = -2
7a) domain all real numbers except 0 and 1
7b) vertical asymptote x = 0 and x = 1
9a) domain all real numbers except 6
9b) answer: vertical asymptote x = 6
11a) domain all real numbers except 3 and -3
11b) vertical asymptote x = 3 and x = -3
13a) domain all real numbers
13b) vertical asymptote none
15a) domain all real numbers
15b) vertical asymptote none
17) x-intercept (3,0)
y-intercept (0,-2)
19) x-intercept (-4,0)
y-intercept (0,-3)
21) x-intercept (-2/3,0) (4,0)
y-intercept (0,2)
23) x-intercept (-4,0) (4,0)
y-intercept none
Section 5.4 continues
25) x-intercept none
y-intercept (0, -1/2)
27) x-intercept (0,0)
y-intercept (0,0)
29) x-intercept (-2,0)
y-intercept (0, 1/8)
31) x-intercept (0,0)
y-intercept (0,0)
33) y = 2
35) y = 0
37) y = 3
41) y = 0
43) y = 0
45) y = 0
51) y = 3x
53) 𝑦 = 2 𝑥
1
49) 𝑦 = 2 𝑥
57) y = x
Section 5.5
1)
a) domain: all real numbers except 6
b) vertical asymptote x = 6
c) horizontal asymptote y = 3
d) x-intercept (-2,0)
e) y-intercept (0,-1)
window used
x-min = -30 x-max = 30
y-min = -30 y-max = 30
1
39) y = 1
5
47) 𝑦 = 4
1
55) 𝑦 = 3 𝑥
Section 5.5
3)
a) domain: all real numbers except 1, -1
b) vertical asymptotes x = 1 x = -1
c) horizontal asymptote y = 4
d) x-intercept (3/2, 0) (-3/2,0)
e) y-intercept (9,0)
window used
x-min -10 x-max 10 y-min -10 y-max 20
5)
a) domain:all real numbers except 0
b) vertical asymptote x = 0
c) horizontal asymptote y = 0
d) x-intercept none
e) y-intercept none
window used x-min -5 x-max 5 ymin -5 y-max 5
Section 5.5
7)
a) domain:all real numbers except 4, -4
b) vertical asymptotes x = 4 x = -4
c) horizontal asymptote y = 0
d) x-intercept (-3,0)
e) y-intercept (0, =3/16)
window x-min -10 x-max 10 y-min -10 ymax 10
9)
a) domain: all real numbers except 5
b) vertical asymptote x = 5
c) horizontal asymptote y = 3x
d) x-intercept (-5/3,0) (-1,0)
e) y-intercept (0,-1)
window used x-min -50 x-max 50 ymin -200 y-max 200
Section 5.5
11)
a) domain: all real numbers except 1/4
b) vertical asymptote x = 1/4
1
c) horizontal asymptote 𝑦 = 𝑥
2
d) x-intercept (0,0)
e) y-intercept (0,0)
window used x-min -10 x-max 10 ymin -10 y-max 10
13)
a) domain: all real numbers except -3
b) vertical asymptote x = -3
c) horizontal asymptote y = 3/2
d) x-intercept (4,0)
e) y-intercept (0,-2)
window used x-min -20 x-max 20 y-min 20 y-max 20
Section 5.5
15)
a) domain: all real numbers except 0
b) vertical asymptote x = 0
c) horizontal asymptote y = 0
d) x-intercept none
e) y-intercept none
window: x-min -10 x-max 10
y-min -10 y-max 10
17)
a) domain: all real numbers except 0
b) vertical asymptote x = 0
c) horizontal asymptote y = 0
d) x-intercept (-5,0)
e) y-intercept none
window used x-min -10 x-max 10 ymin -2 y-max 2
this window shows what happens
around the x-intercept well, but doesn’t
show the vertical asymptote effect very
well. I can’t find a window to show
both.
Section 5.6
1) - 7 < x < 1
3) 1 < x < 5
5) −10 ≤ 𝑥 ≤ 1
7) −1 ≤ 𝑥 ≤ 6
9) x>1 or x < -7
11) x > 5 or x < 1
13) 𝑥 ≥ 1 𝑜𝑟 𝑥 ≤ −10
15) 𝑥 ≥ 1 𝑜𝑟 𝑥 ≤ −6
17) 3 < x < 6
19) -5 < x < 6
21) x > 7 or x < 3
23) x > -1 or x < -5
25) -2 < x < 4
Chapter 5 review
1a) (5,0) even multiplicity, (3,0) odd multiplicity
1b) graph touches x-axis at (5,0) graph crosses x-axis at (3,0)
1c) at most 2 turning points
1d) graph will behave like f(x) = x3 for large values of x.
1e) falls to the left, rises to the right
1f) sketch a graph and approximate the turning points, also label the x-intercepts
y
8
6
4
2
-9
-8
-7
-6
-5
-4
-3
-2
(3.67,1.19)
1 (3,0)
2
3 (5,0)
4
5
-1
x
6
7
8
9
-2
-4
-6
-8
1g) increasing (−∞, 𝟑. 𝟔𝟕) ∪ (𝟓, ∞) decreasing (3.67, 5)
2a) (-4,0) odd mult
(4,0) odd mult
(7,0) odd mult
2b) crosses at (-4,0) crosses at ( 4,0) crosses at ( 7,0)
2c) will have at most 2 turning points.
2d) will behave like f(x) = 3x3
2e) fall left and rises to the right
2f) sketch a graph and approximate the turning points, also label the x-intercepts
y
450
400
(-.95,360)
350
300
250
200
150
100
50
-7
(-4,0)
-5
-4
-6
-3
-2
-1
1
2
-50
(4,0)
3
4
5
(7,0)
6
7
x
8
9
(5.6,-64.5)
2g) increasing (−∞, −. 𝟗𝟓) ∪ (𝟓. 𝟔, ∞) decreasing (-.95, 5.6)
3a) (0,0) has even multiplicity and both (-3,0) (7,0) have odd multiplicity
3b) graph will cross x-axis at (-3,0) and (7,0) and touch at (0,0)
3c) at most 3 turning points.
3d) will behave like f(x) = x4
3e) rises to the left, rises to the right
3f) sketch a graph and approximate the turning points, also label the x-intercepts
y
400
300
200
100
-7
(-3,0)
(0,0)
-4
-3
-2
-1
(-2.07,-36.14)
-6
-5
1
2
3
4
5
(7,0)
6
7
x
8
9
-100
-200
-300
-400
(5.07,-400.36)
3g) 𝒅𝒆𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈 (−∞ − 𝟐. 𝟎𝟕) ∪ (𝟎, 𝟓. 𝟎𝟕) 𝒊𝒏𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈 (−𝟐. 𝟎𝟕, 𝟎) ∪ (𝟓. 𝟎𝟕, ∞)
4a) I graphed and found the graph crossed the x-axis at x = 3. I will do my synthetic division with x =3,
but you could also use -1 or -3
4b) : x3 + x2 – 9x– 9 =(x-3)(x+3)(x+1)
5a) Graph crosses x-axis at x = 1 and x = 2. I will use double synthetic division
5b) x4 +3x3 -8x2 -12x +16 = (x-1)(x-2)(x+2)(x+4)
7) x = -1, ±𝒊√𝟐
6) x = 2,-2,3,-3
8) f(x) = x3 – 4x2 – 2x + 20
9) f(x) = x3 + 5x2 + 16x + 80
10a) all real numbers except 3
10d) (-3,0)
10b) x = 3
10c) y = 2
10e) (0, -2)
10f)
y
8
6
4
2
x
-18
-16
-14
-12
-10
-8
-6
-4
-2
2
(-3,0)
4
6
8
10
12
14
16
18
-2
(0,-2)
-4
-6
-8
11a) all real numbers except -7,3
11b) vertical asymptotes x = -7 and x = 3
11c) y = 0 (the x-axis) 11d) (1,0)
11e) (0, 1/21)
11f)
y
4
3
2
1
x
-14 -13 -12 -11 -10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
-1
-2
-3
-4
12a) all real numbers except 5
12b) vertical asymptote x = 5
12c) y = 2x slant asymptote
12d) (-5/2,0) (-1.0)
12e) (-1,0)
12f)
13) - 8 < x < 2
16) x > 2 or x < -4
14) x > 3 or x<-8
15) 4 < x < 6